Level aspect subconvexity for Rankin-Selberg -functions
Let be a square-free integer and let be a prime not dividing such that with . We prove subconvexity bounds for when and are two primitive holomorphic cusp forms of levels and . These bounds are achieved through an unamplified second moment method.
July 27, 2019
1. Introduction and statement of results
Several authors have recently been successful in implementing the amplification method in order to establish level aspect subconvexity results for Rankin-Selberg convolutions of two forms when one form is fixed and the other form is varying. For example, if is a Hecke cusp form of fixed level and is a Hecke cusp form of varying level , then various bounds of the form
for some absolute positive constant have been shown by Kowalski-Michel-VanderKam , Michel  and Harcos-Michel . Furthermore, results for the Rankin-Selberg convolution of two independently varying forms have been established in the works of Michel-Ramakrishnan , Feigon-Whitehouse  and Nelson  in situations where positivity of the central -values is known. Of particular interest, yet seemingly out of reach by means of current technology, are level and spectral aspect subconvexity results for the Rankin-Selberg convolution of two forms of same level (e.g. when the two forms are same). These -values appear naturally in many areas of number theory and in particular, have important connections with quantum chaos and equidistribution problems.
Subconvexity bounds for an individual -function are often the result of sufficient bounds for a weighted average over an appropriate family of -functions. In this note, we consider the subconvexity problem for the Rankin-Selberg convolution of two varying forms with co-prime levels through the use of a second moment method. With the -function here being constructed from data associated with two independently varying forms, one has a large collection of natural families to choose from.
The ideas presented here may be applied to other Rankin-Selberg convolutions constructed out of multiple independently varying forms. This is the first installment of recent work by the two authors related to the subconvexity problem and its purpose is to demonstrate the existence of situations in which subconvexity may be established through a second moment average without amplification.
We thank IAS Princeton for the wonderful working conditions in which many of the ideas for this collaboration were initially conceived. We also thank MSRI Berkeley, MF Oberwolfach and TIFR Mumbai for providing the opportunity for further discussions. The first author is supported by the Sloan fellowship BR2011-083 and the NSF grant DMS-1068043.
1.1. Holomorphic cusp forms
Let be an integer and be an even integer. We denote by the linear space of holomorphic cusp forms of weight , level and trivial nebentypus. Such forms are holomorphic functions on the upper half-plane satisfying
for every and which vanish at every cusp. Any form has a Fourier series expansion
with coefficients satisfying
as proven by Deligne .
The space is a finite dimensional Hilbert space with respect to the Petersson inner product
We can choose an orthogonal basis for which consists of common eigenfunctions of all the Hecke operators with . That is, each satisfies
for all . Such are called Hecke eigen cusp forms. The Hecke operators are multiplicative and one has that
for any with . In particular, if . Therefore,
if . The Hecke eigenbasis also contains a subset of newforms , those forms which are simultaneous eigenfunctions of all the Hecke operators for any and normalized to have first Fourier coefficient . For , the Hecke relations (3) hold for all integers and it is also known (see ) that
1.2. Rankin-Selberg convolutions of forms with co-prime levels
Let and be two positive square-free co-prime integers and let and be two fixed positive even integers. Given two newforms and , we consider the associated Rankin-Selberg convolution -function (see )
where the and are the local parameters of the -functions associated to and respectively and is the partial Riemann zeta function with the local factors at primes dividing removed. The local parameters satisfy the relations and with the principal character of modulus and similarly for the local parameters associated with . The completed -function is then defined as
The completed -function satisfies the functional equation
The convexity bound for at the point is
for any and may be established in this case simply by the approximate functional equation and Deligne’s bound. It has recently been shown by Heath-Brown , in the general setting of Selberg class -functions using Jensen’s formula for strips, that the in the above bound may be removed
Furthermore, the general results of Soundararajan  provide a “weak-subconvexity” bound of the form
for any .
1.3. Main results
Our purpose here is to provide level aspect subconvexity bounds for the Rankin-Selberg convolution of two forms of varying levels and in situations where both forms are varying at different rates, say for some . The main point we wish to stress, is that we take advantage of the size of the smaller level . The method we present here does not produce subconvexity bounds when nor when is the same size as . Both levels must contribute to the complexity of the problem and they must do so in a manner which is sufficiently distinguishable for the method to work. We restrict to the case of prime to simplify our presentation. Recall that our conductor in this case is of size .
We start by reducing our -function to a smooth sum over Hecke eigenvalues by a standard approximate functional equation argument, see for example , , . Since we are working with newforms of trivial nebentypus, we have
for any positive integer . The derivatives of satisfy
for any . Applying a smooth partition of unity one may derive that (see e.g. )
and is a smooth function, compactly supported on with bounded derivatives and runs over values with .
Since is trivially bounded by for any , the contribution from those is made negligible by choosing above to be sufficiently large. Likewise, if for some , then . Therefore, we are left with
for any . Subconvexity bounds will now follow if one is able to sufficiently bound in the remaining range for . We shall do so by averaging over a Hecke eigenbasis for forms of level .
Theorem 1 (Second Moment).
Let be a positive square-free integer and let be a prime such that . Let and be two fixed positive even integers. Set . Let and choose any . For any new form we have
where the spectral weights are given as .
Note that a second moment bound of the form
for all and any would produce the convexity bound for any individual with and both newforms since then and (see ). Therefore, the bound in Theorem 1 produces a subconvexity bound when with .
Corollary 1 (Subconvexity).
Let be a positive square-free integer and let be a prime not dividing . Let . Let and be two fixed positive even integers. For two newforms and we have
Soften the bound in Theorem 1 to
and equate the second and third terms on the right hand side above while replacing all occurrences of by . ∎
The estimates that we have obtained in Theorem 1 and Corollary 1 are the result of analysis of the shifted convolution sum problem through the -method (, ) with explicit dependence on the level of the form . It is possible to push our arguments further to improve these estimates by considering the shifted convolution sum problem on average over shifts while again maintaining explicit dependence on the level of and we shall do so in a later work. For our purposes here, we prove the following theorem for a fixed non-zero shift.
Theorem 2 (Shifted Convolution Sums).
Let be a non-zero integer and let . Let be a smooth function supported on with partial derivatives satisfying
for some and . For any new forms we have
For other works involving estimates of shifted sums see , , , , , , , , , , , ,  and  for dependence on the level of the forms. The above bound in Theorem 2 does not follow easily from any of the above works. The main advantage here is uniformity with respect to the shift and the coefficient . Furthermore, we note that if then one also has the trivial bound by using (4).
2.1. Bessel functions
We record here some standard facts about the -Bessel functions as can be seen in  as well as several estimates for integrals involving Bessel functions which will be required for our application. One may write the -Bessel functions as
which, when is a positive integer, one has that
Using the above facts leads us to the following results.
Let be integers and let . Define
where is a smooth function compactly supported on with bounded derivatives. We have
for any .
A change of variables, , gives
Therefore, we see from (6) that may be written as the sum of four similar terms, one of them being
Repeated integration by parts gives the desired result. ∎
For as in Lemma 1, we have
Differentiate and use the bound in (8). ∎
Let be positive integers with and let be a non-zero integer. Take and . For any , define
where is the function from Lemma 7 in §2.2 and is a smooth function supported on with partial derivatives satisfying
for some and . We have
for any non-negative integers and . Furthermore,
A change of variables, integrating by parts once in and applying the given bounds for the functions and the Bessel functions gives
Repeating the argument, for instead of , gives the bound (11). ∎
2.2. Summation Formulae, Large Sieve and the -method
Let be an integer. For any , let denote the Kloosterman sum
The Kloosterman sums satisfy the Weil bound
where is the number of divisors of . This bound is best possible for an individual Kloosterman sum. Sums of Kloosterman sums appear in the following spectral average (see  for a derivation).
Lemma 4 (Petersson trace formula).
Let be an integer. Let be any Hecke eigenbasis for . For any , we have
where the spectral weights are given by
and if and otherwise.
One also has the following large sieve estimate.
Let be a smooth function supported on such that for all . For any sequences of complex numbers we have
with any . Moreover the exponent may be replaced by .
The above estimate will be useful in controlling the size of Kloosterman sum moduli. For all remaining moduli we will apply the following analogue to Poisson summation.
Lemma 6 (Voronoi summation,  Theorem A.4).
Let and let be a smooth function, compactly supported in . Let be a holomorphic newform of level and weight . Set . Then there exists a complex number of modulus (depending on and ) and a newform of the same level and the same weight such that
where denotes the multiplicative inverse of .
We will now briefly recall a version of the circle method introduced in  and . The starting point is a smooth approximation of the -symbol. We will follow the exposition of Heath-Brown in .
For any there is a positive constant , and a smooth function defined on , such that
The constant satisfies for any . Moreover for all , and is non-zero only for .
In practice, to detect the equation for a sequence of integers in the range , it is logical to choose . The smooth function satisfies (see )
for and . Also for , we have
3. Initial reduction of the second moment
Let be a positive square-free integer and let be a prime not dividing . Let and be two positive even fixed integers. Fix a newform and choose an orthogonal Hecke eigenbasis for . Set . Let and choose any . As seen in the statement of Theorem 1, we are interested in obtaining upper bounds for the sum
where and is smooth, compactly supported on with bounded derivatives. We start by opening the square and applying the Petersson trace formula in . Since is a newform, we have and so
The “diagonal term” satisfies
for any . This is the first term seen in the bound in Theorem 1. We are now left with the “off-diagonal” terms
We start by truncating the sum over . By the Weil bound for individual Kloosterman sums and bounds for the Bessel functions in §2.1, there exist positive values and such that the sum over may be truncated to those up to an error term of size at most . For the remaining sum over , we introduce another smooth partition of unity and break the sum into dyadic segments of size , as we did with our -sum above, so that we are left with sums of type
where is a smooth function supported on . Note that must be of size at least by the congruence condition. Furthermore, an application of Lemma 5 shows that
which is smaller than the bound in Theorem 1 as soon as . Therefore, bounding the second moment in (15) has reduced to the following statement.
Let . For any we have
where is given by (16) above and runs over dyadic values.
With additional work, one might also eliminate all , for some depending on , in order to improve the final range of sizes relative to for which subconvexity is achieved. To keep our presentation short, we shall only show how one may remove (see Lemma 10).
We emphasize here the significance of the level in our problem. Note that the first term in the above bound (17), which came from the diagonal term after applying the Petersson trace formula in , beats the convexity bound for by . If were fixed, then Lemma 8 would already be insufficient for subconvexity.
4. Reduction to Shifted Convolution Sums
Let . We now proceed with the analysis of , as defined by (16) above, when with . Opening the Kloosterman sums and changing the order of summation, one is left to study
As in the works ,  and , an application of Voronoi summation in and the evaluation of the resulting Ramanujan sums will lead to a collection of shifted convolution sums. Switching from Kloosterman sums to Ramanujan sums in such a manner was already seen in the work of Goldfeld . Since the application of Voronoi summation will be for a newform of level and therefore depends on the divisibility of by , we first break apart our sum as
Voronoi summation in then gives that the inner sum, up to a constant, is equal to
This produces a Ramanujan sum over for each modulus , which we write as
Summing over will now produce a congruence condition between and modulo . Thus, we have reduced (18) to the following.
Let and let be as in (16) with . For any we have
with shifted convolution sums
In the above, determines the main contribution in the sum over and which occurs when and . The other ranges of summation are negligible as can be seen by Lemma 1.
5. Proof of Theorem 1
Theorem 1 will follow after an appropriate treatment of the shifted convolution sums in Lemma 9. We break this apart into cases according to the value of .
5.1. Treatment of the shifted sums
Since we are dealing with forms of level prime, we only have two types of shifted convolution sums to consider, those with and those with . In the latter case, the moduli must be of size at least by the congruence condition. Applying Lemma 1 and the bound obtained from Lemma 2 one has that
so that this contribution to bounding is
which matches the first term in (17).